A different direction to speeding up digital quantum simulation is to exploit information about the initial state. The error of digital quantum simulation is commonly quantified in previous work by the spectral-norm distance, which considers all possible states in the underlying Hilbert space. But if the state is known to be within some subspace throughout the simulation, then in principle this knowledge could be used to improve the algorithm. For instance, digital quantum simulation in practice often starts with an initial state in the low-energy subspace of the Hamiltonian, so a worst-case spectral-norm analysis will inevitably overestimate the error. To address this, recent studies have considered a low-energy projection on the simulation error and provided improved approaches, using either Trotterization [1]}, [2]}, [3]}, [4]} or more advanced quantum algorithms [5]}, that can be advantageous when the energy of the initial state is sufficiently small.
